Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1004.4911 · Journal of Mathematical Physics

On the efficiency of Hamiltonian-based quantum computation for low-rank matrices

Zhenwei Cao, Alexander Elgart

Published Mar 1, 2012• 6 citations
arXivHTMLPDF

Abstract

We present an extension of adiabatic quantum computing (AQC) algorithm for the unstructured search to the case when the number of marked items is unknown. The algorithm maintains the optimal Grover speedup and includes a small counting subroutine. Our other results include a lower bound on the amount of time needed to perform a general Hamiltonian-based quantum search, a lower bound on the evolution time needed to perform a search that is valid in the presence of control error and a generic upper bound on the minimum eigenvalue gap for evolutions. In particular, we demonstrate that quantum speedup for the unstructured search using AQC type algorithms may only be achieved under very rigid control precision requirements.

DOI
10.1063/1.3690045

Related papers

On the efficiency of Hamiltonian–based quantum computation for low–rank matrices

Zhenwei Cao, Alexander Elgart

We present an extension of Adiabatic Quantum Computing (AQC) algorithm for the unstructured search to the case when the number of marked items is unknown. The algorithm maintains the optimal Grover speedup and includes a small counting subroutine. Our other results include a lower bound on the amount of time needed to perform a general Hamiltonian-based quantum search, a lower bound on the evolution time needed to perform a search that is valid in the presence of control error and a generic upper bound on the minimum eigenvalue gap for evolutions. In particular, we demonstrate that quantum speedup for the unstructured search using AQC type algorithms may only be achieved under very rigid control precision requirements.

•Jan 1, 2012•6 citations

Adiabatic quantum computation: Enthusiast and Sceptic's perspectives

Zhenwei Cao, Alexander Elgart

Enthusiast's perspective: We analyze the effectiveness of AQC for a small rank problem Hamiltonian $H_F$ with the arbitrary initial Hamiltonian $H_I$. We prove that for the generic $H_I$ the running time cannot be smaller than $O(\sqrt N)$, where $N$ is a dimension of the Hilbert space. We also construct an explicit $H_I$ for which the running time is indeed $O(\sqrt N)$. Our algorithm can be used to solve the unstructured search problem with the unknown number of marked items. Sceptic's perspective: We show that for a robust device, the running time for such $H_F$ cannot be much smaller than $O(N/\ln N)$.

arXiv:1004.5409•Apr 29, 2010•0 citations

Unstructured Adiabatic Quantum Optimization: Optimality with Limitations

Arthur Braida, Shantanav Chakraborty, Alapan Chaudhuri +4 more

In the circuit model of quantum computing, amplitude amplification techniques can be used to find solutions to NP-hard problems defined on n -bits in time poly(n)2n/2 . In this work, we investigate whether such general statements can be made for adiabatic quantum optimization, as provable results regarding its performance are mostly unknown. Although a lower bound of Ω(2n/2) has existed in such a setting for over a decade, a purely adiabatic algorithm with this running time has been absent. We show that adiabatic quantum optimization using an unstructured search approach results in a running time that matches this lower bound (up to a polylogarithmic factor) for a broad class of classical local spin Hamiltonians. For this, it is necessary to bound the spectral gap throughout the adiabatic evolution and compute beforehand the position of the avoided crossing with sufficient precision so as to adapt the adiabatic schedule accordingly. However, we show that the position of the avoided crossing is approximately given by a quantity that depends on the degeneracies and inverse gaps of the problem Hamiltonian and is NP-hard to compute even within a low additive precision. Furthermore, computing it exactly (or nearly exactly) is #P-hard. Our work indicates a possible limitation of adiabatic quantum optimization algorithms, leaving open the question of whether provable Grover-like speed-ups can be obtained for any optimization problem using this approach.

arXiv:2411.05736•Quantum•Jul 11, 2025•2 citations

Adiabatic quantum unstructured search in parallel

Sean A. Adamson, Petros Wallden

We present an optimized adiabatic quantum schedule for unstructured search building on the original approach of Roland and Cerf [Phys. Rev. A 65, 042308 (2002)]. Our schedule adiabatically varies the Hamiltonian even more rapidly at the endpoints of its evolution, preserving Grover's well-known quadratic quantum speedup. In the errorless adiabatic limit, the probability of successfully obtaining the marked state from a measurement increases directly proportional to time, suggesting efficient parallelization. Numerical simulations of an appropriate reduced two-dimensional Schrödinger system confirm adiabaticity while demonstrating superior performance in terms of probability compared to existing adiabatic algorithms and Grover's algorithm, benefiting applications with possible premature termination. We introduce a protocol that ensures a marked-state probability at least $p$ in time of order $\sqrt{N}(1+p/\varepsilon)$, and analyze its implications for realistic bounded-resource scenarios. Our findings suggest that quantum advantage may still be achievable under constrained coherence times (where other algorithms fail), provided the hardware allows for them to be sufficiently long.

arXiv:2502.08594•Feb 12, 2025•0 citations

Quantum search by local adiabatic evolution

Jérémie Roland, Nicolas J. Cerf

The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state, which tends to the solution. We apply this time-dependent Hamiltonian approach to Grover's problem, i.e., searching a marked item in an unstructured database. We find that by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order $\sqrt{N},$ where N is the number of items in the database. We thus recover the advantage of Grover's standard algorithm as compared to a classical search, scaling as N. This is in contrast with the constant-rate adiabatic approach of Farhi et al. (e-print quant-ph/0001106), where the requirement of adiabaticity is expressed only globally, resulting in a time of order N.

arXiv:quant-ph/0107015•Physical Review A•Mar 26, 2002•629 citations

How fast can quantum annealers count?

Itay Hen

We outline an algorithm for the quantum counting problem using adiabatic quantum computation (AQC). We show that the mechanism of quantum-adiabatic evolution may be utilized toward estimating the number of solutions to a problem, and not only to find them. Using local adiabatic evolution, a process in which the adiabatic procedure is performed at a variable rate, the problem of counting the number of marked items in an unstructured database is solved quadratically faster than the corresponding classical algorithm. The above algorithm provides further evidence for the potentially powerful capabilities of AQC as a paradigm for more efficient problem solving on a quantum computer, and may be used as the basis for solving more sophisticated problems.

arXiv:1301.4956•Journal of Physics A Mathematical and Theoretical•May 27, 2014•11 citations
We present an extension of adiabatic quantum computing (AQC) algorithm for the unstructured search to the case when the number of marked items is unknown. The algorithm maintains the optimal Grover speedup and includes a small counting subroutine. Our other results include a lower bound on the amount of time needed to perform a general Hamiltonian-based quantum search, a lower bound on the evolution time needed to perform a search that is valid in the presence of control error and a generic upper bound on the minimum eigenvalue gap for evolutions. In particular, we demonstrate that quantum speedup for the unstructured search using AQC type algorithms may only be achieved under very rigid control precision requirements.
We present an extension of Adiabatic Quantum Computing (AQC) algorithm for the unstructured search to the case when the number of marked items is unknown. The algorithm maintains the optimal Grover speedup and includes a small counting subroutine. Our other results include a lower bound on the amount of time needed to perform a general Hamiltonian-based quantum search, a lower bound on the evolution time needed to perform a search that is valid in the presence of control error and a generic upper bound on the minimum eigenvalue gap for evolutions. In particular, we demonstrate that quantum speedup for the unstructured search using AQC type algorithms may only be achieved under very rigid control precision requirements.
Enthusiast's perspective: We analyze the effectiveness of AQC for a small rank problem Hamiltonian HFH_FHF​ with the arbitrary initial Hamiltonian HIH_IHI​. We prove that for the generic HIH_IHI​ the running time cannot be smaller than O(N)O(\sqrt N)O(N​), where NNN is a dimension of the Hilbert space. We also construct an explicit HIH_IHI​ for which the running time is indeed O(N)O(\sqrt N)O(N​). Our algorithm can be used to solve the unstructured search problem with the unknown number of marked items. Sceptic's perspective: We show that for a robust device, the running time for such HFH_FHF​ cannot be much smaller than O(N/ln⁡N)O(N/\ln N)O(N/lnN).
In the circuit model of quantum computing, amplitude amplification techniques can be used to find solutions to NP-hard problems defined on n -bits in time poly(n)2n/2 . In this work, we investigate whether such general statements can be made for adiabatic quantum optimization, as provable results regarding its performance are mostly unknown. Although a lower bound of Ω(2n/2) has existed in such a setting for over a decade, a purely adiabatic algorithm with this running time has been absent. We show that adiabatic quantum optimization using an unstructured search approach results in a running time that matches this lower bound (up to a polylogarithmic factor) for a broad class of classical local spin Hamiltonians. For this, it is necessary to bound the spectral gap throughout the adiabatic evolution and compute beforehand the position of the avoided crossing with sufficient precision so as to adapt the adiabatic schedule accordingly. However, we show that the position of the avoided crossing is approximately given by a quantity that depends on the degeneracies and inverse gaps of the problem Hamiltonian and is NP-hard to compute even within a low additive precision. Furthermore, computing it exactly (or nearly exactly) is #P-hard. Our work indicates a possible limitation of adiabatic quantum optimization algorithms, leaving open the question of whether provable Grover-like speed-ups can be obtained for any optimization problem using this approach.
We present an optimized adiabatic quantum schedule for unstructured search building on the original approach of Roland and Cerf [Phys. Rev. A 65, 042308 (2002)]. Our schedule adiabatically varies the Hamiltonian even more rapidly at the endpoints of its evolution, preserving Grover's well-known quadratic quantum speedup. In the errorless adiabatic limit, the probability of successfully obtaining the marked state from a measurement increases directly proportional to time, suggesting efficient parallelization. Numerical simulations of an appropriate reduced two-dimensional Schrödinger system confirm adiabaticity while demonstrating superior performance in terms of probability compared to existing adiabatic algorithms and Grover's algorithm, benefiting applications with possible premature termination. We introduce a protocol that ensures a marked-state probability at least ppp in time of order N(1+p/ε)\sqrt{N}(1+p/\varepsilon)N​(1+p/ε), and analyze its implications for realistic bounded-resource scenarios. Our findings suggest that quantum advantage may still be achievable under constrained coherence times (where other algorithms fail), provided the hardware allows for them to be sufficiently long.
The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state, which tends to the solution. We apply this time-dependent Hamiltonian approach to Grover's problem, i.e., searching a marked item in an unstructured database. We find that by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order N,\sqrt{N},N​, where N is the number of items in the database. We thus recover the advantage of Grover's standard algorithm as compared to a classical search, scaling as N. This is in contrast with the constant-rate adiabatic approach of Farhi et al. (e-print quant-ph/0001106), where the requirement of adiabaticity is expressed only globally, resulting in a time of order N.
We outline an algorithm for the quantum counting problem using adiabatic quantum computation (AQC). We show that the mechanism of quantum-adiabatic evolution may be utilized toward estimating the number of solutions to a problem, and not only to find them. Using local adiabatic evolution, a process in which the adiabatic procedure is performed at a variable rate, the problem of counting the number of marked items in an unstructured database is solved quadratically faster than the corresponding classical algorithm. The above algorithm provides further evidence for the potentially powerful capabilities of AQC as a paradigm for more efficient problem solving on a quantum computer, and may be used as the basis for solving more sophisticated problems.